Let $M$ be an $m$ dimensional submanifold in the Euclidean space
${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain
some low geometric estimates of the total square mean curvature
$\int_M H^2 d\sigma$. The low bounds are geometric invariants
involving the volume of $M$, the total scalar curvature of $M$,
the Euler characteristic and the circumscribed ball of $M$.

We investigate the geometry of manifolds with bounded Ricci
curvature in $L^1$-sense. In particular, we generalize the
classical volume comparison theorem to our situation and obtain a
generalized sphere theorem.